# The Kalman condition for the boundary controllability of coupled 1-d wave equations

## TL;DR

This paper establishes a necessary and sufficient Kalman condition for the boundary controllability of coupled 1D wave equations with distinct eigenvalues, providing a comprehensive controllability criterion and describing the attainable set.

## Contribution

It introduces a Kalman condition for boundary controllability of coupled wave equations with distinct eigenvalues, advancing theoretical understanding of controllability criteria.

## Key findings

- Kalman condition is necessary and sufficient for controllability
- Controllability depends on the eigenvalues of the coupling matrix
- The attainable set is characterized but not necessarily optimally

## Abstract

This paper is devoted to prove the exact controllability of a system of N one-dimensional coupled wave equations when the control is exerted on a part of the boundary by means of one control. We consider the case where the coupling matrix A has distinct eigenvalues. We give a Kalman condition (necessary and sufficient) and give a description, non-optimal in general, of the attainable set.